# of a function

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##### 1: 27.13 Functions
Jacobi (1829) notes that $r_{2}\left(n\right)$ is the coefficient of $x^{n}$ in the square of the theta function $\vartheta\left(x\right)$: …(In §20.2(i), $\vartheta\left(x\right)$ is denoted by $\theta_{3}\left(0,x\right)$.) … Mordell (1917) notes that $r_{k}\left(n\right)$ is the coefficient of $x^{n}$ in the power-series expansion of the $k$th power of the series for $\vartheta\left(x\right)$. …
##### 5: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
10.46.3 $E_{a,b}\left(z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(ak+b\right)},$ $a>0$.
This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to\infty$, together with a smooth interpretation of Stokes phenomena. …
##### 6: 8.17 Incomplete Beta Functions
where, as in §5.12, $\mathrm{B}\left(a,b\right)$ denotes the beta function: …
8.17.4 $I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right).$
For the hypergeometric function $F\left(a,b;c;z\right)$ see §15.2(i). …
8.17.13 $(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),$
8.17.20 $I_{x}\left(a,b\right)=I_{x}\left(a+1,b\right)+\frac{x^{a}(x^{\prime})^{b}}{a% \mathrm{B}\left(a,b\right)},$
##### 7: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). … To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau\left(n\right)$, and the values can be checked by the congruence (27.14.20). …
##### 8: Peter L. Walker
Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
##### 9: 21.8 Abelian Functions
An Abelian function is a $2g$-fold periodic, meromorphic function of $g$ complex variables. …For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 10: Mourad E. H. Ismail
Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special FunctionsA volume dedicated to Mizan Rahman (with E. …